2.2 Scientiﬁc Notation & Dimensional Analysis

2.2 Scientific Notation & Dimensional Analysis

Monday, September 23, 13

Scientific Notation

• Can be used to express any number as a number between 1 and 10 (coefficient) multiplied by 10 raised to any power (exponent).

• 36,000 = 3.6 x 104

• Positive exponent = how much the coefficient must be multiplied by 10.

• 0.00036 = 3.6 x 10-4


Scientific Notation

• The value of the exponent = the number of places the decimal point moved.

• Exponent is positive when decimal point moves to the left.

• EXAMPLES

• Exponent is negative when the decimal point moves to the right.

• EXAMPLES


Math with Scientific Notation

•Addition and Subtraction

• Exponents must be the same.

• EXAMPLES

• If not the same, rewrite with same exponent.

• EXAMPLES


Scientific Notation

•Multiplication & Division

• Two-step process.

• Multiply / divide coefficients.

• Add / subtract exponents

• EXAMPLES


Dimensional Analysis• Use conversion factors to to convert one unit to

another.

• Conversion factor is a ratio of equivalent values having different units.

• Relationships between units

• EXAMPLES


Dimensional Analysis

• Use conversion factors to to convert one unit to another.

• Conversion factor is a ratio of equivalent values having different units.

• Prefixes are the source of many conversion factorsPrefix Symb

olNumerical

ValueConversion Factor

Mega M 1,000,000 1 Mg = 1,000,000 g

Kilo K 1,000 1 Kg = 1,000 g

Deci d 0.1 10 dg = 1 g

Centi c 0.01 100 cg = 1 g

Milli m 0.001 1000 mg = 1 g

Micro u 0.000001 1,000,000 ug = 1 gMonday, September 23, 13

Using Conversion Factors

• Must accomplish two things:

• Must cancel one unit.

• Must introduce a new one.

• All units except what you want must cancel out.

• EXAMPLE

• How many pizzas do I need to order to give each person in class two pieces? (1 pizza = 8 pieces)


2.3 Uncertainty in Data


Uncertainty in Data

• Every measurement contains some amount of error.

• Must evaluate both accuracy and precision each time.


Uncertainty in Data

• Accuracy

• The closeness of a measured value to an accepted value.

• Precision

• The closeness a series of measurements are to one another.


Uncertainty in Data

2.3 Uncertainty in Data

Accuracy

Precision

6


Error

Percent Error

22. The density of copper is 8.92g/cm3. During a lab, you calculate the density ofcopper to be 8.66g/cm3. What is your % error?

7

Uncertainty in Data

๏Which student is the most accurate?๏Student A

๏Which student is the most precise?๏Student C


Error and Percent Error

• Accuracy of experiment is measured by comparing how close the experimental value comes to the accepted value.

• Experimental value

• Measured during an experiment.

• Accepted value

• True value


Error and Percent Error

• Experimental Error

• Difference between experimental and accepted values.

• Error = experimental value - accepted value

• Percent Error

• Expresses error as percentage of accepted value.

• Percent Error = | error | X 100

accepted value

Accepted value= 1.59 Error = 0.05 Percent Error = 3.14%


2.4 Significant Figures


Significant Figures

• Precision is limited by the tools available.

• Indicated by the number of digits reported

• Significant figures include all known digits plus one estimated digit.


Significant FiguresSignificant Figures

Rules for Significant Figures

1. All non zero numbers ARE significant2. All zeros at the beginning of a number are NOT significant3. All zeroes in between non zero numbers ARE significant4. Zeroes at the end MAY OR MAY NOT be significant

If there is a decimal point anywhere in the problem, it is significantIf NO decimal point exists it is NOT significant.

Significant Figure Practice

How many significant figures does each number have

23. .00353

24. 59302

25. 50.3

26. 3400

27. .034040

28. 370

8

What is the measurement of the rod reported in significant figures?








23. .00353

24. 59302

25. 50.3

26. 3400

27. .034040

28. 370

8

What is the measurement of the rod reported in significant figures?








23. .00353

24. 59302

25. 50.3

26. 3400

27. .034040

28. 370

8


Significant Figures

Significant Figures






23. .00353

24. 59302

25. 50.3

26. 3400

27. .034040

28. 370

8

1) 508.0 = 4 2) 820,400.0 = 7 3) 807,000 = 3

4) 0.049450 = 5 6) 0.00084 = 2

Rules for significant figures

1. Zeroes at the beginning of a number are NOT significant. (example: .000349)

2. Zeroes at the end of a number without a decimal point in the number are NOT

significant. (example: 65000)

3. All nonzero numbers ARE significant. (45.69)

4. All zeroes in between nonzero number ARE significant. (example: 305)

5. Final zeroes to the RIGHT of a decimal point ARE significant. (example:

.0034900)

6. Counting numbers and defined constants have infinite number of significant

figures. (example: 6 molecules or 60 s = 1 min)

Practice – how many significant figures does each number have?

1. 84.00

2. .00649

3. 2600

4. 2504.60

5. .0094020

6. 9400.

Rules for addition and subtraction

The answer should have the same number of decimal places as the original value

with the least number of decimal places (example: 3.54 + 5.9 = 9.4)

Practice – complete each addition or subtraction problem and round the answer to

the correct number of decimal places.

7. 79.469 + 40.2

8. 94.33 – 12.4987

9. .1148 + .94624

Rules for multiplication and division

The answer should have the same number of significant figures as the original

value with the least number of significant figures (example: 3.0 × 12.0 = 36)

Practice -‐ complete each multiplication or division problem and round the answer

to the correct number of significant figures.


Significant Figures








.0034900)




1. 84.00

2. .00649

3. 2600

4. 2504.60

5. .0094020

6. 9400.






7. 79.469 + 40.2

8. 94.33 – 12.4987

9. .1148 + .94624













.0034900)




1. 84.00

2. .00649

3. 2600

4. 2504.60

5. .0094020

6. 9400.






7. 79.469 + 40.2

8. 94.33 – 12.4987

9. .1148 + .94624






= 119.7= 81.83= 1.0610


Significant Figures








.0034900)




1. 84.00

2. .00649

3. 2600

4. 2504.60

5. .0094020

6. 9400.






7. 79.469 + 40.2

8. 94.33 – 12.4987

9. .1148 + .94624













.0034900)




1. 84.00

2. .00649

3. 2600

4. 2504.60

5. .0094020

6. 9400.






7. 79.469 + 40.2

8. 94.33 – 12.4987

9. .1148 + .94624






10. 4.6 × 13.5

11. .0049 × 3.9987

12. 5.90 / .36

13. 46.50 / .049

14. 3.75 × 20

= 62.1 = 62= 0.01959 = 0.020= 16.389 = 16


Significant Figures

• Rounding Numbers

• Identify the last significant figure.

• If number to right of last significant figure is:

• Less than Five, DO NOT change: 35.3 = 35

• Greater than FIVE, round UP: 35.8 = 36

• Five:

• Do nothing if the last significant figure is EVEN: 3.525014 = 3.52

• Round UP if the last significant figure is ODD: 3.515014 = 3.52


2.4 Representing Data


Representing Data

• “A picture is worth.....

• A graph is a picture

• Scientists use graphs to present data in a form that allows them analyze results and communicate information about their experiments.


Graphing

• Goal of experiments is to discover patterns within situations.

• Does changing temp change rate of rxn?

• Does change in diet affect rat’s ability to navigate maze?

• Data listed in tables may not make patterns obvious.

• Using data to create graphs can help reveal patterns.

• Graph = visual display of data.


Circle Graphs (Pie Chart)

• Useful for showing parts of a fixed whole.

• Comparison of data

• Parts usually labeled as percents.



Calculations with significant figures

Multiplication and division rules:

29. .037 × 34.55

30. 98.3 ÷ .30

Addition and Subtraction rules:

31. 12.34 + 9.852

32. 39.0 - 89.481


Pie or Circle Graph

9



Calculations with significant figures

Multiplication and division rules:

29. .037 × 34.55

30. 98.3 ÷ .30

Addition and Subtraction rules:

31. 12.34 + 9.852

32. 39.0 - 89.481


Pie or Circle Graph

9


Bar Graphs

• Shows how quantities vary across categories.

• Include time, location, and temperature.

• Quantity being measured goes on y-axis.

• Which variable is this?

• Quantity that a scientist changes goes on x-axis.

• Which variable is this?


Bar Graphs Bar Graph -

Line Graph -

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Line Graphs

• Type of graph most used in chemistry.

• Points represent the intersection of of data for two variables.

• Like a bar graph: independent variable = x-axis

dependent variable = y-axis


Line Graphs • Points represent the intersection of of data for

two variables.

• Like a bar graph: independent variable = x-axis

dependent variable = y-axis

Bar Graph -

Line Graph -

10Monday, September 23, 13

Line Graphs • Contain best-fit line.

• Line drawn such that equal number of points fall above and below line.

Bar Graph -

Line Graph -

10Monday, September 23, 13

Line Graphs • Contain best-fit line.

• Line drawn such that equal number of points fall above and below line.

Bar Graph -

Line Graph -

10


Line Graphs • If the best-fit line is straight, there is a linear

relationship between variables.

• Directly related

• If the best-fit line is curved, there is a nonlinear relationship between variables.

• Inverse relationship


Line Graphs

• Slope of line tells HOW variables are related.

• Slope equation =


Line Graphs

• Slope of line tells HOW variables are related.

• Rising slope = positive slope.

• Both dependent and independent variable increase.

• Sinking slope = negative slope.

• Dependent variable decreases as independent variable increases.


Line Graphs • Slope of line tells HOW variables are related.

Bar Graph -

Line Graph -

10

Bar Graph -

Line Graph -

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How to Interpret Graphs

• Determine independent and dependent variables.

• x-axis vs y-axis?

• Decide if relationship is linear or nonlinear.

• If linear, determine if slope is positive or negative.


Interpolation vs Extrapolation

• Connected points on line graph = continuous data.

• Reading values between recorded data points is called interpolation.



• Connected points on line graph = continuous data.

• Reading values between recorded data points is called interpolation.

Bar Graph -

Line Graph -

10

What is the temperature at an elevation of 350 m?



• Extending line beyond plotted points to estimate values for variables = extrapolation.

• Can lead to errors and inaccuracy.



• Extending line beyond plotted points to estimate values for variables = extrapolation.

• Can lead to errors and inaccuracy.

Bar Graph -

Line Graph -

10

What is the temperature at an elevation of 800 m?


2.2 Scientiﬁc Notation & Dimensional Analysis

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